Spinning three-dimensional regular black hole solutions in extended Einstein theory

We investigate rotating black hole configurations in three-dimensional vacuum f(R) gravity as an extension of standard Einstein theory. Exact solutions are obtained and shown to depend on the mass M, angular momentum J, and an effective cosmological parameter Λ, even though no explicit cosmological constant is included in the action. Unlike the BTZ solution of General Relativity, the resulting geometries exhibit a non-constant Ricci scalar, reflecting the modified gravitational dynamics. We demonstrate that the standard rotating BTZ black hole is recovered as a limiting case when f(R) = R. In contrast, genuinely modified solutions arise for non-linear forms of f(R), including configurations that remain regular, with curvature invariants free of divergences at both small and large radial distances. However, other solutions display distinct asymptotic behavior, where curvature invariants signal strong singularities at spatial infinity, in clear contrast with the constant invariants of the BTZ spacetime. Furthermore, we reconstruct the corresponding f(R) functions and show that they admit polynomial forms. The physical viability of the solutions is examined through thermodynamic and dynamical considerations. In particular, stability is supported by the positivity of the heat capacity and by satisfying the Ostrogradsky stability condition, namely f″(R) > 0.